Question 103
In an acute-angled triangle $ABC$, let $H$ be the foot of the altitude from vertex $A$. Let $P$ be a moving point in the plane such that the internal bisectors of the angles $\angle PBC$ and $\angle PCB$, denoted $k$ and $\ell$ respectively, intersect at a point lying on the segment $AH$. Let $k$ meet $AC$ at $E$, let $\ell$ meet $AB$ at $F$, and let $EF$ meet $AH$ at $Q$. Prove that, however $P$ moves (subject to the above condition), the line $PQ$ always passes through a fixed point.